2009/04/02 by Jean-Loup Waldspurger, Waldspurger, Jean-Loup
Mathematics · #22E50 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:22E50
paper · pdf · doi:10.48550/arxiv.0904.0314
arxiv created 2009/04/02 · arxiv updated 2009/12/01
Let F be a non-archimedean local field, of characteristic 0. Let V be a finite dimensional vector space over F and q be a non-degenerate quadratic form on V. Denote G the special orthogonal group of (V,q). Let W a non-degenerate hyperplane of V, denote H the special orthogonal group of W. Let π, resp. σ, an admissible irreducible representation of G(F), resp. H(F). Denote m(σ,π) the dimension of the complex space HomH(F)(π| H(F),σ). It's know that m(σ,π)=0 or 1. In a first paper, we have defined another term mgeom(σ,π). It's an explicit sum of integrals of functions that can be deduced from the characters of σ and π. Assume that π and σ are tempered. Then we prove the equality m(σ,π)=mgeom(σ,π). This generalize the result of the first paper, where π was supercuspidal. As in this paper, the previous equality implies as corollary (assuming certain properties of tempered L-packets) a weak form of the local Gross-Prasad conjecture, now for pairs of tempered L-packets.